Invisible knots and rainbow rings: knots not determined by their determinants

Abstract

We determine p-colorability of the paradromic rings. These rings arise by generalizing the well-known experiment of bisecting a Mobius strip. Instead of joining the ends with a single half twist, use $m$ twists, and, rather than bisecting ($n = 2$), cut the strip into $n$ sections. We call the resulting collection of thin strips $P(m,n)$. By replacing each thin strip with its midline, we think of $P(m,n)$ as a link, that is, a collection of circles in space. Using the notion of $p$-colorability from knot theory, we determine, for each $m$ and $n$, which primes $p$ can be used to color $P(m,n)$. Amazingly, almost all admit 0, 1, or an infinite number of prime colorings! This is reminiscent of solutions sets in linear algebra. Indeed, the problem quickly turns into a study of the eigenvalues of a large, nearly diagonal matrix. Our paper combines this explicit calculation in linear algebra with a survey of several ideas from knot theory including colorability and torus links.